We focus on the portion of the floor that is left unpainted. After one day, $ \frac12 $ of the floor is left unpainted. After two days, $ \frac1{2^2} $ of the floor is left unpainted, and so on. After $ n $ days, $ \frac1{2^n} $ of the floor is unpainted. The floor has area $ 15^2 = 225 $ square feet, so we look for the least number of days $ n $ when at most $ \frac1{225} $ of the floor is left unpainted: \begin{align*} \frac1{2^n} &\leq \frac1{225}\\ \Rightarrow 2^n &\geq 225\\ \Rightarrow n&\geq8.\end{align*} Thus, Zeno took $ \boxed{8} $ days to paint the floor.